Continuity and Differentiability

Let$f\left(x\right)=\left|2x^{2}+5\left|x\right|-3\right|,x\in\,R$.If $m$ and $n$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $m+n$ is equal to:

Let$f:R\rightarrow\,R$be a function defined by$f\left(x\right)=\max\,\left{x,x^{2}\right}$.Let$S$denote the set of all points in$R$,where$f$is not differentiable.Then :

Let $f:\left[0,\infty\,\right)\rightarrow\,\left[0,3\right]$ be a function defined by$f\left(x\right)=\left{\begin{matrix} \max\,\left{\sin\,t:0\leq\,t\leq\,\pi\,\right},x\in\,\left[0,\pi\,\right] \\ 2+\cos\,x,x>\pi\,…

Let $f:R\rightarrow\,R$ be defined as$f\left(x\right)=\left{\begin{matrix} x^{5}\sin\,\left(\frac{1}{x}\right)+5x^{2} & , & x0 \end{matrix}\right$. The value of $\lambda\,$ for which $f^{"}(0)$ exists, is___.

The number of points, at which the function $f\left(x\right)=\left|2x+1\right|-3\left|x+2\right|+\left|x^{2}+x-2\right|,x\in\,R$ is not differentiable, is

Let $S={\left(\lambda\,,\mu\,\right)\in\,R\times\,R:f\left(t\right)=\left(\left|\lambda\,\right|e^{\left|t\right|}-\mu\,\right)\sin\,\left(2\left|t\right|\right),t\in\,R$is a differential function}. Then,$S$is a subset…

Let a function $f:R\rightarrow\,R$ be defined as,$f\left(x\right)=\left{\begin{matrix} \sin\,x-e^{x} & \mathrm{if}x\leq\,0 \\ a+\left[-x\right] & \mathrm{if}0Where $$\left[x\right]$$ is the greatest integer less than or…

Let the functions $f:R\rightarrow\,R$ and $g:R\rightarrow\,R$ be defined as : $f\left(x\right)=\left{\begin{matrix} x+2, & xThen, the number of points in $R$ where $$\left(fog\right)\left(x\right)$ is NOT differentiable…

Let $a,b\in\,R,b\neq\,0$. Defined a function,$f\left(x\right)=\left{\begin{matrix} a\sin\,\frac{\pi\,}{2}(x-1),\mathrm{for}x\leq\,0 \\ \frac{\tan\,2x-\sin\,2x}{bx^{3}},\mathrm{for}x>0 \end{matrix}\right$ If $f$ is…

Let $f:R\rightarrow\,R$ be defined as$f\left(x\right)=\left{\begin{matrix} \frac{x^{3}}{(1-\cos\,2x)^{2}}\log\,_{e}\left(\frac{1+2xe^{-2x}}{\left(1-xe^{-x}\right)^{2}}\right) & ,x\neq\,0 \\ \alpha\, & ,x=0…

Passage: Let $f(x)=\tan(x^2)$ and $g(x)=x|x|$ for $|x|Question: If $p(x)=f(x)g(x)$, then which of the following statements is/are correct ? I. $p(x)$ is continuous at $x=0$. II. $p(x)$ is differentiable at $x=0$. Select…

Passage: Let $f(x)=\tan(x^2)$ and $g(x)=x|x|$ for $|x|Question: If $$q(x)=f\circ$ g(x)$, then which of the following statements is/are correct ? I. $q(x)$ is continuous at $x=0$. II. $q(x)$ is differentiable at $x=0$.…